Without any other hypothesis than that of its electric charge, the electron is found to have inertia defined as capacity for kinetic energy, but with a particular law of variation of this as a function of the velocity, and this inertia appears to approach infinity as the velocity approaches that of light.
The behavior of this law depends very little on the hypothesis made as to the form of the electron and the distribution of the electric charge which it carries. In all cases it is found to be impossible to give the electron a velocity equal to that of light, at least permanently.
Instead of considering with Max Abraham the electron to be spherical at all velocities, Lorentz admits it to be spherical when at rest and to have a uniform distribution of charge; but if all internal forces are solely electromagnetic or act as such, we have the view that the electron is flattened in the direction of motion by a quantity proportional to the square of the ratio (β=v/V). of its velocity to that of light, becoming an ellipsoid of revolution, the equatorial diameter remaining equal to that of the original. This leads, as we shall see, to a law of inertia different from that of an invariable sphere.
We shall likewise see that it does not appear to be necessary to assign to the electrons, the negative ones at least, any other inertia than this in order to account for the dynamic properties of the cathode rays; however, experiments are not yet sufficiently exact to allow us to infer the form of the electron itself, which depends on the law of the variation of the kinetic energy with the velocity.
(14) Two Problems. We have examined, so far, only the case of an electron in uniform motion in the absence of any external electromagnetic field capable of modifying the motion of the electron by giving it an acceleration.
The general problem of the connection between the ether and the electron, which probably represents the most important of the connections between ether and matter, is double.
In the first place, what is the electromagnetic disturbance in the ether accompanying any given motion of the electrons whatsoever?
In the second place, what motions would free electrons have if displaced in an external magnetic field superimposed on that which constitutes their wake?
(15) The Velocity Wave - The Acceleration Wave. We actually possess all the elements necessary for the solution of the first problem, in which the motion is uniform in a particular case. Lorentz has given in a very simple form the general solution by the use of a delayed potential.
Each element of the charge in motion is determined by its position, its velocity, and its acceleration at the time T, the electric and magnetic fields at the time T + t, on a sphere having for its centre the
